Construction of L²-orthogonal elements of arbitrary order for Local Projection Stabilization

We construct L²-orthogonal conforming elements of arbitrary order for the Local Projection Stabilization (LPS). L²-orthogonal basis functions lead to a diagonal mass matrix which can be advantageous for time discretizations. We prove that the constructed family of finite elements satisfies a local inf-sup condition. Additionally, we investigate the size of the local inf-sup constant with respect to the polynomial degree. Our numerical tests show that the discrete solution is oscillation-free and of optimal accuracy in the regions away from the boundary or interior layers.